# A Hot Shower

Recently, and rather unexpectedly, I purchased a new hot water heater. I discovered I was about to make this purchase in the shower, as I suspect is the usual way.

After the hot water heater was installed, I took the opportunity to adjust the temperature of the new hot water heater to hotter than “default,” but not all the way to “your skin will melt off.” My hope was that the upstairs shower — which had always gotten “hot enough” but only when the dial was rotated all the way to the left — would now have a “too hot” setting that could be “rotated to the right.”

Happily, the plan worked, but a new problem emerged.

Let’s say the minimum temperature of my shower is T_{min} and the maximum temperature of the shower is T_{max}. If the shower dial can be rotated 180°, then each degree of shower dial rotation corresponds 1/180th of the range of temperatures:

${T}_{\mathrm{max}}-{T}_{\mathrm{min}}=\mathrm{Range\; of\; possible\; temperatures}$

When I increased T_{max} at the hot water heater, that range of possible temperatures got wider — and as such, every degree of shower dial rotation now means a corresponding larger change in temperature.

Plugging in some real numbers, if the old hot water heater was set to 120°F (Tmax), and the minimum temperature that came out of the wall was 50°F:

${T}_{\mathrm{max}}=\mathrm{120\xb0F}$ ${T}_{\mathrm{min}}=\mathrm{50\xb0F}$

… then previously every degree of rotation of the shower dial provided:

$\frac{\mathrm{120\xb0F}-\mathrm{50\xb0F}}{\mathrm{180\xb0\; rotation}}=\mathrm{0.39\xb0F\; of\; temperature\; change\; per\; degree\; rotated}$

If the new hot water heater is set at the higher 140°F, then every degree of dial rotation now corresponds to:

$\frac{\mathrm{140\xb0F}-\mathrm{50\xb0F}}{\mathrm{180\xb0\; rotation}}=\mathrm{0.50\xb0F\; of\; temperature\; change\; per\; degree\; rotated}$

While it doesn’t seem that an additional 0.11°F per 1° of rotation (0.50°F - 0.39°F) could *possibly* make much difference, I assure you it is now much more challenging to find a comfortable shower temperature.

Clearly I’ve been thinking about this, while fiddling with the dial in the shower, quite a bit since my purchase.

Today, in a moment of serendipity, I bumped into a manifesto on shower temperature control from Ben Holmen. In it he discovers, and laments, that of all the possible positions on the shower dial only around 10% result in comfortable shower temperatures. He has the graphs to back it up too.

It seems that $500 solutions to this problem exist, because of course they do, but at that price a little extra adjusting of the dial doesn’t seem quite so bad.